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2. Below is a list V₂ of vector spaces over R. You are given subsets UiC Vi. Decide which of these are subspaces. Justify your answers by giving a proof or a counter-example in each case. (1) V₁ = R¹ and U₁ = { (ao, a₁, a2, a3) € R¹ | ₁² a₁ =0}. ai (2) V₂ = R³ and U₂ = { (a, b, c) € R³ | ab = c}.

2. Below is a list V₂ of vector spaces over R. You are given subsets UiC Vi. Decide which of these are subspaces. Justify your answers by giving a proof or a counter-example in each case. (1) V₁ = R¹ and U₁ = { (ao, a₁, a2, a3) € R¹ | ₁² a₁ =0}. ai (2) V₂ = R³ and U₂ = { (a, b, c) € R³ | ab = c}.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 26EQ
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could you please help me with 1 and 2 and can you please provide explanations

2. Below is a list V₂ of vector spaces over R. You are given subsets U₁ V₁. Decide which of these are subspaces. Justify your answers by giving a proof or a counter-example in each case. (1) V₁ = R¹ and U₁ = { (ao, a₁, A2, A3) € R4 | Σ²_o a; = 0}. i=0 (2) V₂ = R³ and U₂ = { (a, b, c) € R³ | ab = c }. (3) V3 R³ and U3 = {(a + b, b, a+b): a, b ≤ R}. (4) V₁ = R³ and U₁ = {(a, b, c) € R³: a+b+c= 0 and b + c = 0}. (5) V5 = R³ and U5 = {(a, b, c) € R³ : a = 0 or b = 0 or c = 0}. (6) V6 R[x] is the vector space of polynomials with real coefficients (this was example 3 in the lecture). U6 = {p E R[x] : p(1)=0}. (7) V₂ = C([0, 1]) is the vector space of continuous functions from the interval [0, 1] to R. U+= {f e C([0, 1]) : fồf(z) - xdz =0}.
Transcribed Image Text:2. Below is a list V₂ of vector spaces over R. You are given subsets U₁ V₁. Decide which of these are subspaces. Justify your answers by giving a proof or a counter-example in each case. (1) V₁ = R¹ and U₁ = { (ao, a₁, A2, A3) € R4 | Σ²_o a; = 0}. i=0 (2) V₂ = R³ and U₂ = { (a, b, c) € R³ | ab = c }. (3) V3 R³ and U3 = {(a + b, b, a+b): a, b ≤ R}. (4) V₁ = R³ and U₁ = {(a, b, c) € R³: a+b+c= 0 and b + c = 0}. (5) V5 = R³ and U5 = {(a, b, c) € R³ : a = 0 or b = 0 or c = 0}. (6) V6 R[x] is the vector space of polynomials with real coefficients (this was example 3 in the lecture). U6 = {p E R[x] : p(1)=0}. (7) V₂ = C([0, 1]) is the vector space of continuous functions from the interval [0, 1] to R. U+= {f e C([0, 1]) : fồf(z) - xdz =0}.
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